Secant lines: challenging problem 2

We're asked, at which points on the graph is f of x times f prime of x equal to 0? So if I have the product of two things and it's equal to 0, that tells us that at least one of these two things need to be equal to 0. So first of all, let's see are there any points when f of x is equal to 0? So we're plotting f of x on the vertical axis. We could call this graph right over here, we could say this is y is equal to f of x. So at any point, does the y value of this curve equal 0? So it's positive, positive, positive, positive, positive, positive. But it is decreasing right over here. Well, it's decreasing here. Then it's increasing. Then it's decreasing. And it does get to 0 right over here, but that's not one of the labeled points. And they want us to pick one of the labeled points or maybe even more than one of these labeled points. So we're going to focus on where f prime of x is equal to 0. And we just have to remind ourselves what f prime of x even represents. f prime of x represents the slope of the tangent line at that value of x. So for example, f prime of 0-- which is the x value for this point right over here-- is going to be some negative value. It's the slope of the tangent line. Similarly, f prime of x, when x is equal to 4-- that's what's going on right over here-- that's going to be the slope of the tangent line. That's going to be a positive value. So if you look at all of these, where is the slope of the tangent line 0? And what does a 0 slope look like? Well, it looks like a horizontal line. So where is the slope of the tangent line here horizontal? Well, the only one that jumps out at me is point B right over here. It looks like the slope of the tangent line would indeed be horizontal right over here. Or another way you could think of it is the instantaneous rate of change of the function, right at x equals 2, looks like it's pretty close to-- if this is x equals 2-- looks like it's pretty close to 0. So out of all of the choices here, I would say only B looks like the derivative at x equals 2. Or the slope of the tangent line at B, it looks like it's 0. So I'll say B right over here. And then they had this kind of crazy, wacky expression here. f of x minus 6 over x. What is that greatest in value? And we have to interpret this. We have to think about what does f of x minus 6 over x actually mean? Whenever I see expressions like this, especially if I'm taking a differential calculus class, I would say well, this looks kind of like finding the slope of a secant line. In fact, all of what we know about derivatives is finding the limiting value of the slope of a secant line. And this looks kind of like that, especially if at some point, my y value is a 6 here. And this could be the change in y value. And if the corresponding x value is 0, then this would be f of x minus 6 over x minus 0. So do I have 0, 6 on this curve here? Well, sure. When x is equal to 0, we see that f of x is equal to 6. So what this is right over here-- let me rewrite this. This we could rewrite as f of x minus 6 over x minus 0. So what is this? What does this represent? Well, this is equal to the slope-- let me do some of that color-- this is equal to the slope of the secant line between the points, x, f of x, x, and whatever the corresponding f of x is. And we could write it as 0, f of 0 because we see f of 0 is equal to 6. This right over here is f of 0. In fact, let me just write that as 6. And the point, 0, 6. So let's go through each of these points and think about what the slope of the secant line between those points are and point A. This is essentially the slope of the secant line between some point x, f of x, and essentially point A. So let's draw this out. So between A and B you have a fairly negative slope. Remember we want to find the largest slope. So here it's fairly negative. Between A and C, it's less negative. Between A and D, it's even less negative. It's still negative, but it's less negative. And then between A and E, it becomes more negative now. And then between A and F, it becomes even more negative. So when is the slope of the secant line between one of these points and A the greatest? Or I guess we could say the least negative? Because it seems like they're always negative. It would be between point D and A. So when is this greatest in value? Well, when we're looking at point D. At point D, x is equal to 6 and it looks like f of x is like 5 and 1/2 or something. So this will turn into f of 6, which is 5 and 1/2 or maybe it's even less than that-- 5 and 1/3 or something, minus 6 over 6 minus 0. That's how we'll maximize this value. This is the least negative slope of the secant line.